In this fourth part, $A \in \mathcal{S}_n(\mathbb{R})$ is a symmetric matrix whose eigenvalues are denoted $\lambda_1 \leqslant \lambda_2 \leqslant \cdots \leqslant \lambda_n$. We consider an arbitrary orthonormal basis $\left(\mathbf{u}_1, \ldots, \mathbf{u}_n\right)$. Let $\mathbf{U}$ be a random variable taking values in the finite set $\left\{\mathbf{u}_1, \ldots, \mathbf{u}_n\right\}$, following the uniform distribution on this set. Show that for all $\mathbf{w} \in \mathbb{R}^n$, we have $\mathbb{E}\left[\langle \mathbf{U}, \mathbf{w} \rangle^2\right] = \frac{1}{n} \|\mathbf{w}\|^2$.
In this fourth part, $A \in \mathcal{S}_n(\mathbb{R})$ is a symmetric matrix whose eigenvalues are denoted $\lambda_1 \leqslant \lambda_2 \leqslant \cdots \leqslant \lambda_n$. We consider an arbitrary orthonormal basis $\left(\mathbf{u}_1, \ldots, \mathbf{u}_n\right)$. Let $\mathbf{U}$ be a random variable taking values in the finite set $\left\{\mathbf{u}_1, \ldots, \mathbf{u}_n\right\}$, following the uniform distribution on this set. Show that for all $\mathbf{w} \in \mathbb{R}^n$, we have $\mathbb{E}\left[\langle \mathbf{U}, \mathbf{w} \rangle^2\right] = \frac{1}{n} \|\mathbf{w}\|^2$.